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in the theory of functions of a complex variable
Classical results of N.N. Luzin and I.I. Privalov that clarify the character of a boundary uniqueness property of analytic functions (cf. Uniqueness properties of analytic functions) (see [1]).
1) Let be a meromorphic function of the complex variable
in a simply-connected domain
with rectifiable boundary
. If
takes angular boundary values zero on a set
of positive Lebesgue measure on
, then
in
. There is no function meromorphic in
that has infinite angular boundary values on a set
of positive measure.
2) Let be a meromorphic function in the unit disc
other than a constant and having radial boundary values (finite or infinite) on a set
situated on an arc
of the unit circle
that is metrically dense and of the second Baire category (cf. Baire classes) on
. Then the set
of its radial boundary values on
contains at least two distinct points. Metric density of
on
means that every portion of
on
has positive measure. This implies that if the radial boundary values of
on a set
of the given type are equal to zero, then
in
. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set
of the given type.
Luzin and Privalov (see [1], [2]) constructed examples to show that neither metric density nor second Baire category are by themselves sufficient for the assertion in 2 to hold.
See also Boundary properties of analytic functions; Luzin examples; Cluster set; Privalov theorem; Riesz theorem.
References
[1] | N.N. [N.N. Luzin] Lusin, I.I. [I.I. Privalov] Priwaloff, "Sur l'unicité et la multiplicité des fonctions analytiques" Ann. Sci. Ecole Norm. Sup. (3) , 42 (1925) pp. 143–191 |
[2] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[3] | A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian) |
Comments
References
[a1] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 |
Luzin-Privalov theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin-Privalov_theorems&oldid=12447